Electrically-augmented damping

ABSTRACT

Provided herein are devices, systems, and methods for electrically-augmented damping of an actuator and associated devices. In particular, electrically-augmented damping derived from measurement of voltage across an actuator and current flowing through an actuator is provided.

CROSS-REFERENCE TO RELATED APPLICATION

The present invention claims priority to U.S. Provisional PatentApplication Ser. No. 61/421,486 filed Dec. 9, 2010, which is herebyincorporated by reference in its entirety.

FIELD

Provided herein are devices, systems, and methods forelectrically-augmented damping of an actuator and associated devices. Inparticular, electrically-augmented damping derived from measurement ofvoltage across an actuator and current flowing through an actuator isprovided.

BACKGROUND

Fourier transform interferometric spectrometers are widely used in theanalysis of chemical compounds. By measuring the absorption of radiationby an unknown sample at various wave lengths and comparing the resultswith known standards, these instruments generate useful information withrespect to the chemical makeup of the unknown sample.

Conventional Fourier-transform (FT) spectrometers are based on Michelsoninterferometer arrangements. In a typical FT spectrometer, light from alight emitting source is collected, passed through an interferometer andsample to be analyzed, and is brought to focus on a detector (Saptari,Vidi, Fourier-Transform Spectroscopy Instrumentation Engineering. SPIEPress, Bellingham Wash., Vol. No.: TT61 (2004), herein incorporated byreference in its entirety). The interferometer system, in combinationwith the sample, modulates the intensity of the light that strikes thedetector, and thereby forms a time variant intensity signal. A detector,analog-to-digital converter, and processor then receive, convert, andanalyze the signal.

In an FT spectrometer utilizing an interferometer (SEE FIG. 1), inputlight is divided into two beams by a beam splitter. One beam isreflected off a fixed mirror and one off a moving mirror. While the pathlength of the beam striking the fixed mirror is substantially constant,movement of the moving mirror alters the path length of the other beam,thereby changing the distance the beam travels in comparison to thereference beam (e.g., the beam reflected off the fixed mirror. Thischanging path length introduces a time delay between the beams uponreaching the detector. The time-offset beams interfere or reinforce eachother, allowing the temporal coherence of the light to be measured atdifferent time delay settings, effectively converting the time domaininto a spatial coordinate. Measurements of the signal at many discretepositions of the moving mirror are used to construct a spectrum using aFT of the temporal coherence of the light. The power spectrum of the FTof the interferogram corresponds to the spectral distribution of theinput light. The moving mirror allows a time-domain interferogram to begenerated which, when analyzed, allows high resolution frequency-domainspectra to be produced. A Fourier transform is performed on the data toproduce a spectrum which shows spectral-energy versus frequency.

It is critical in the design of these instruments that the surface ofthe moving mirror be very accurately held in an orthogonal position,both to the fixed mirror and to the direction of the motion of themoving mirror. Mirror positional accuracy is of importance becausedeviations in the mirror alignment produce small errors in thetime-domain interferogram which may translate into large errors in thefrequency-domain spectrum. In a typical interferometer, mirrordeviations larger than one wave length of the analytical radiation areconsidered significant and can degrade the quality of the entireinstrument. Contemporary high- and moderate-performanceFourier-transform spectrometry instruments may utilize stabilizationassemblies, typically flexure assemblies, to support the moving mirrorin the interest of minimizing mechanical hysteresis and other non-lineareffects.

Despite all measures taken to reduce undesired alteration of the movingmirror's position, some degree of unwanted or unintended deviation mayoccur in even high quality systems. The motion of the mirror in responseto applied force is described by a second-order differential equationwhose coefficients are determined by the mass of the moving mirror, theeffective spring constant of the flexures, the frictional lossintroduced by suspension, and viscous loss introduced by motion throughthe atmosphere. Forces directed at the moving mirror may be a result ofa command sent to the mirror actuator (e.g., a voice-coil actuator) aspart of the normal function of the FTS instrument, or may be undesireddisturbances, caused, for example, by incidental motion of the entireinstrument. The frictional losses present in the flexure suspension arevery low, a condition which leads to a lightly-damped system response toapplied force. Such a system is significantly more difficult to controlthan one with a greater level of inherent damping. The lightly-dampedsystem is also more susceptible to externally-produced disturbanceforces than one with greater damping.

SUMMARY

In some embodiments, provided herein are methods of providingelectrically-augmented damping of a voice-coil actuator comprising:subtracting a rate-feedback voltage from an actuator command voltage,wherein the rate feedback voltage is derived from measurement of voltageacross the actuator and current flowing through the actuator. In someembodiments, the voltage across the actuator comprises a component dueto the flow of current through impedances of the actuator, and acomponent due to movement of an armature. In some embodiments, therate-feedback voltage is derived by subtracting a voltage equal to thecomponent of the voltage across the actuator due to the flow of currentthrough the impedances of the actuator from the voltage across theactuator. In some embodiments, the voltage is equal to the component ofthe voltage across the actuator due to the flow of current through theimpedances of the actuator is derived by scaling the voltage appearingat terminal β (the bottom of the actuator). In some embodiments, thevoltage appearing at terminal β (the bottom of the actuator) is only afunction of the current flowing through the actuator. In someembodiments, the voltage appearing at terminal β (the bottom of theactuator) is substantially a function only of the current flowingthrough the actuator. In some embodiments, the electrically-augmenteddamping results in reduced performance anomalies for the voice-coilactuator. In some embodiments, the rate-feedback voltage is calculatedwithout use of a velocity sensor.

In some embodiments, provided herein are systems and devices comprisingcircuitry for providing electrically-augmented damping of a voice-coilactuator, comprising: (a) a transconductance amplifer A1, wherein anoutput of the transconductance amplifier A1 is a function of: (i)command voltage, and (ii) a rate-feedback voltage; (b) an actuator,wherein the actuator comprises: (i) a permanent magnet, and (ii) anarmature that carries current through a magnetic field associated withthe permanent magnet; (c) an amplifier A2, wherein an output of theamplifier A2 is a function of the voltage appearing at terminal β (thebottom of the actuator); and (d) an amplifier A3, wherein an output ofthe amplifier A3 is a function of: (i) the voltage appearing at terminalα (the top of the actuator), and (ii) a voltage proportional to thecomponent of the voltage appearing at terminal α (the top of theactuator) due to the flow of current through the actuator. In someembodiments, the output of the transconductance amplifier A1 flows intothe actuator. In some embodiments, the voltage appearing at terminal β(the bottom of the actuator) is a function of flow of current throughthe actuator. In some embodiments, the voltage appearing at terminal α(the top of the actuator) is a function of: (i) flow of current throughthe actuator, and (ii) movement of the armature through the magneticfield associated with the permanent magnet. In some embodiments, theamplifier A2 scales and inverts the voltage appearing at terminal β (thebottom of the actuator) to produce a voltage equal in magnitude to, but180 degrees out of phase with, the component of the voltage acrossappearing at terminal α (the top of the actuator) due to the flow ofcurrent through the actuator. In some embodiments, the output ofamplifier A3 is the sum of the output of amplifier A2 and the voltageappearing at terminal α (the top of the actuator). In some embodiments,the output of amplifier A3 is equal to the voltage resulting from themovement of the armature through the magnetic field associated with thepermanent magnet. In some embodiments, the output of amplifier A3 is therate-feedback voltage.

In some embodiments, provided herein are voice-coil actuators associatedwith a system for electrically-augmented damping. In some embodiments, arate-feedback voltage is subtracted from an actuator command voltage toeffectively damp the voice-coil actuator. In some embodiments, the ratefeedback voltage is derived from measurement of voltage across theactuator and current flowing through the actuator. In some embodiments,the electrically-augmented damping results in reduced performanceanomalies for the voice-coil actuator. In some embodiments, therate-feedback voltage is calculated without use of a velocity sensor.

In some embodiments, provided herein are Fourier-transform spectrometerscomprising an optical path-length modulator utilizing a voice-coilactuator and comprising a system for electrically-augmented damping. Insome embodiments, a rate-feedback voltage is subtracted from theactuator command voltage to effectively damp the voice-coil actuator. Insome embodiments, the rate feedback voltage is derived from measurementof voltage across the actuator and current flowing through the actuator.In some embodiments, the rate-feedback voltage is calculated without useof a velocity sensor. In some embodiments, the electrically-augmenteddamping results in reduced performance anomalies for the voice-coilactuator. In some embodiments, reduced performance anomalies for thevoice-coil actuator provide a reduction in errors in time-domaininterferograms, a reduction in errors in frequency-domain spectrum, andimproved spectrometer performance.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing summary and detailed description is better understood whenread in conjunction with the accompanying drawings which are included byway of example and not by way of limitation.

FIG. 1 shows a schematic of a Fourier-transform spectrometer (adaptedfrom: Saptari, Vidi, Fourier-Transform Spectroscopy InstrumentationEngineering. SPIE Press, Bellingham Wash., Vol. No.: TT61 (2004), hereinincorporated by reference in its entirety).

FIG. 2 shows a circuit schematic employed in embodiments in accordancewith the present disclosure.

FIG. 3 shows a sample plot of compensated plant loop gain.

FIG. 4 shows a sample plot of compensated plant closed-loop gain.

FIG. 5 shows a schematic of a circuit sub-component employed inadditional embodiments.

FIG. 6 shows a sample plot of a Transconductance (Gm) amplifieropen-loop transfer function.

FIG. 7 shows a sample plot of a Gm amplifier closed-loop transferfunction.

FIG. 8 shows a schematic of a circuit sub-component employed inadditional embodiments.

FIG. 9 shows a sample plot of a Phase-Lead (lead) amplifier closed-loopgain.

FIG. 10 shows a sample plot of lead amplifier peak output voltages for alocked armature.

FIG. 11 shows a schematic of a Summing Amplifier circuit sub-componentemployed in some embodiments.

FIG. 12 shows a plot of SNR of rate feedback signal.

FIG. 13 shows a plot of transfer function of synthesized circuits.

DEFINITIONS

When used to compare two or more voltages, currents, or other values,the term “equal,” indicates that the voltages or currents aresubstantially identical, or identical within reasonable error associatedwith the circuitry or system. Such reasonable errors are understood bythose in the art, and may be, for example <5% error, <2% error, <1%error, <0.1% error, <0.01% error, <0.001% error, <0.0001% error, orless, depending upon the particular utility. A skilled artisan iscapable of determining such limits. The values need not be preciselyidentical; however, values which are merely similar or approximate donot satisfy the criteria of being “equal.”

As used herein, the terms “actuator” and “motor” are used synonymously.

As used herein, the terms “top of the actuator” and “terminal α” areused synonymously.

As used herein, the terms “bottom of the actuator” and “terminal β” areused synonymously.

DETAILED DESCRIPTION OF EMBODIMENTS

Provided herein are devices, systems, computer-executable instructions,and methods for electrically-augmented damping of an actuator andassociated devices. Example actuators include, but are not limited to,voice-coil actuators. In some embodiments, methods for increasedeffective damping without the use of a dedicated velocity sensor areprovided. In some embodiments, a rate signal is developed (e.g., for usein damping) without input from a dedicated velocity sensor. In someembodiments, electrically-augmented damping derived from measurement ofvoltage across an actuator and current flowing through an actuator isprovided. In particular, a rate signal is developed, for use in damping,from measurements of the voltage appearing between the two terminals ofthe voice-coil actuator and the current flowing through the actuator.Embodiments are described in which a dedicated velocity sensor is notused, thereby reducing cost and volume of the system.

Provided herein are methods for providing a compensation voltage forelectrically-augmented damping of any suitable system. In someembodiments, the compensation voltage is calculated, determined, and/orderived from currents and voltages within the system. In someembodiments, currents and voltages within the system are manipulated inany suitable manner to provide a compensation voltage (e.g., ratefeedback) to damp the system. In some embodiments, a compensationvoltage is provided by rate feedback circuitry or compensationcircuitry. In some embodiments, a compensation voltage or rate-feedbacksignal is provided without additional devices or apparatuses (e.g.,velocity sensors). In some embodiments, a compensation voltage orrate-feedback signal is provided without additional devices orapparatuses (e.g., velocity sensors) other than compensation circuitryto obtain, divert, and manipulate currents and voltages within theexisting circuitry.

Provided herein are devices, systems, and methods forelectrically-augmented damping of actuators which find use in anysuitable device, system, or apparatus. In embodiments, a computingdevice executing a program of instructions may be utilized to performthe relevant calculation in order to control the system and/or performthe method. As is to be appreciated, the instructions may be embodied ina variety of media, including, but not limited to, tangible media suchas hard drives, memory (e.g., random access memory, read only memory,magnetic and optic media and so forth. In some embodiments, voice-coilactuators find use in, for example, Fourier-transform spectrometers(e.g., FT infrared spectrometers), hard disk drives, loudspeakers,shaker tables, lens focusing, medical equipment, laser-cutting tools,etc. In some embodiments, an actuator utilizing the damping describedherein is an element in an optical path-length modulator mechanism of aFourier-transform spectrometer. In some embodiments, Fourier-transformspectrometers comprising an optical path-length modulator mechanismemploying electrically-augmented damping are provided. In someembodiments, increased effective damping results in noise reduction orincreased precision for devices and systems utilizing systems andmethods described herein. In particular, increased effective damping inthe optical path-length modulator mechanism of a Fourier-transformspectrometer results in a reduction in errors in time-domaininterferograms. These errors may translate into errors (e.g.,comparatively larger errors) in the frequency-domain spectrum; hence,reducing such errors may results in improved spectrometer performancewhile minimizing the overall “footprint” of such a device. Similarperformance enhancements are realized in other devices and systemsutilizing the damping methods and systems described herein.

In some embodiments, electronically-augmented damping actuators (e.g.,non-linear actuators, voice-coil actuators, rotational actuators, etc.)are provided. In some embodiments, a voice-coil actuator is provided. Insome embodiments, a voice-coil actuator is provided for use in aFourier-transform spectrometer, loudspeaker, hard disk drive, shakertables, lens focusing, medical equipment, laser-cutting tools, etc. Insome embodiments, a voice-coil actuator is employed in an opticalpath-length modulator mechanism of a Fourier-transform spectrometer(e.g., providing movement of the “moving mirror”). In some embodiments,a voice-coil actuator is an electromagnetic device that producesaccurately controllable forces over a limited stroke with a single coilor phase (Goque & Stepak, Voice-coil actuators, G2 Consulting,Beaverton, Oreg., herein incorporated by reference in its entirety). Insome embodiments, a voice-coil actuator is a linear actuator. In someembodiments, a voice-coil actuator is a swing-arm actuator or rotationactuator, and is used to rotate a load through an angle. In someembodiments, a voice-coil actuator is capable of accelerations greaterthan 10 times gravitational acceleration (e.g., >10 g . . . 15 g . . .20 g . . . 25 g . . . 30 g . . . 40 g . . . 50 g, etc.). In someembodiments, a voice-coil actuator is capable of precise positioning(e.g., error of less than one hundred thousandth of an inch (e.g.,<0.00001, <0.000001, <0.0000001, <0.00000001, etc.). In someembodiments, a voice-coil actuator has a settling time, e.g., the timerequired for structural vibration to settle down to below themeasurement threshold after a high-acceleration move, of less than 10milliseconds (e.g., <10 ms, <5 ms, <2 ms, <1 ms, <0.5 ms, <0.1 ms, <0.01ms, etc.). A voice-coil actuator functions based on the principle thatif a conductor (e.g., wire) carrying electric current passes through amagnetic field, a force is generated on the conductor orthogonal to boththe direction of the current and the magnetic flux. Likewise, if acurrent is passed through a conductor which lies between two poles of amagnetic field, a force will be generated on the conductor orthogonal toboth the direction of the current and the magnetic flux. The ratio ofgenerated force to current may be referred to as the “force constant”.The generated force varies as a function of the amount of current,direction of current, coil speed, coil position, rate of change of thecurrent, strength of magnetic field, etc. Once the conductor (e.g.,armature) begins to move as a result of the generated force, a voltageis induced in the conductor, caused by its motion in a magnetic field.This voltage, called back electromotive force (EMF), is proportional tospeed, field strength and current, and is in a direction to opposemotion. It reduces the voltage across the coil, lowering the current andthe rate of acceleration.

In some embodiments, a voice-coil actuator comprises a permanent magnet,and an armature that carries a current through the magnetic field of thepermanent magnet. In some embodiments, a command voltage is applied tothe voice-coil actuator. In some embodiments, a force is induced on thearmature due to the flow of current through the magnetic field. In someembodiments, back EMF is produced by movement of the armature, herebyreducing the voltage across the voice-coil. In some embodiments, systemsand methods are provided to compensate for the voltage caused by themovement of the armature. In some embodiments, methods and systemsherein serve to alter the effective command voltage applied to avoice-coil actuator to compensate for undesired inputs (e.g., voltagecaused by armature movement, noise, fluctuations, etc.). In someembodiments, a compensation voltage is applied to the command voltage.In some embodiments, a compensation voltage is a rate signal. In someembodiments, a compensation voltage is analogous to a rate signal. Insome embodiments, a compensation voltage is derived from a rate signal.In some embodiments, an effective command voltage is produced byapplying a compensation voltage to a command voltage. In someembodiments, a voltage analogous to the armature rate (e.g.,compensation voltage) is removed from the command voltage applied to thevoice-coil actuator. In some embodiments, a voltage analogous to thearmature rate (e.g., compensation voltage) is removed from the commandvoltage to yield an effective command voltage. In some embodiments, arate signal is developed, for use in damping, from measurements of thevoltage appearing between the two terminals of the conductor thevoice-coil actuator and the current flowing through the actuator. Insome embodiments, a compensation voltage is derived from the voltageacross the voice-coil actuator and the current flowing through theactuator. In some embodiments, a compensation voltage is derived fromsumming the voltage appearing at the top of the actuator and a voltageequal to the inverse of the component of voltage appearing at the top ofthe actuator produced by the flow of current through the actuator.

FIG. 2 provides a schematic representing exemplary circuitry forapplying the methods provided herein. In some embodiments, a series ofamplifiers and/or circuits are employed to obtain a compensation voltageto correct the command voltage producing an effective command voltage.

Amplifier A1 is a transconductance (G_(m)) amplifier (SEE FIG. 2). Insome embodiments, the transconductance amplifier A1 is configuredaccording to the circuit synthesis outlined in FIG. 5. Other suitabletransconductance amplifier circuitry can be employed as well. The outputof transconductance amplifier A1 is a current which is a function of:(1) the command voltage e_(Cmd) and (2) the compensation voltage orrate-feedback voltage e_(δ). In some embodiments, the output oftransconductance amplifier A1 is a current which is a function of thesum of the command voltage and the compensation voltage.

In some embodiments, the output current produced by transconductanceamplifier A1 flows into the actuator, dividing between the motorelectrical impedance Z_(m) and the shunt resistance R_(Sh) (SEE FIG. 2).In some embodiments, the total current from the Z_(m) and R_(Sh) thenflows through Sense Resistance R_(Sense). The voltage generator depictedin the motor model of FIG. 2 represents the voltage produced by themotion of the armature through the magnetic field associated with thepermanent magnet. The magnitude of this voltage is given by the productof the angular velocity of the armature (Θ-dot) and the Back-EMFconstant of the motor (K_(BEMF)). This generator is represented in thecomplex-frequency domain model with amplitude s*Θ*K_(BEMF). The voltageappearing at the top of the motor e_(α) contains (1) a component due tothe flow of current through the motor impedances e_(α1), and (2) acomponent due to armature motion e_(α2). The voltage appearing at thebottom of the motor e_(β) is a function of current flowing through theactuator (e.g., no component due to armature movement). Other motors,generators, and actuators that produce voltages according to otherschemes are provided herein. For example, other suitable motors are usedin place of the one described in FIG. 2. Changes to the circuitry can beused to adapt to different motor setups.

The lead amplifier A2 may scale the voltage observed at the bottom ofthe motor e_(β) such that the output of the lead amplifier A2 issubstantially identical to that component of the voltage appearing atthe top of the motor which is due to motor current (SEE FIG. 2). Thelead amplifier A2 can be configured according to the circuit synthesisoutlined in FIG. 8. In some embodiments, the transfer function of leadamplifier A2 scales the voltage e_(β) such that lead amplifier A2 outpute_(γ) is equal to the component of e_(α) produced by the flow of currentthrough the motor impedances. In this way the lead amplifier A2 caninvert the phase of the current. In some embodiments, lead amplifier A2introduces a phase shift of 180 degrees, thereby inverting the phase ofthe current.

In some embodiments, the rate summer amplifier A3 adds voltages e_(γ)and e_(α) (SEE FIG. 2), due to the inversion produced by lead amplifierA2. This results in a subtraction. In some embodiments, the voltage atthe output of rate summer amplifier A3 is the compensation voltagee_(δ), which is equal to the motional voltage s*Θ*K_(BEMF) scaled by thegain of rate summer amplifier A3. The rate summer amplifier A3 may beconfigured according to the circuit synthesis outlined in FIG. 11. Theoutput of the rate summer amplifier A3 represents only the motionalvoltage produced by armature motion, the effects of motor current havingcancelled in the subtraction. Thus, the compensation voltage e_(δ) isanalogous to the velocity at the motor armature, Θ-dot.

In some embodiments, the compensation voltage e_(δ) is subtracted fromthe command voltage applied to transconductance amplifier A1 to yield aneffective command voltage (SEE FIG. 2). As e_(δ) is analogous toarmature rate, the compensation (e.g., voltage subtraction) augments thedamping inherent in the physical mechanism. In some embodiments, thevoltage compensation at transconductance amplifier A1 accounts forvoltage effects of the movement of the armature through the magneticfield.

The circuitry and/or amplifier connectivity described above should notbe viewed as limiting. Additional embodiments, although not explicitlydescribed herein are contemplated based on design preference and can beused with the described embodiments. Certain embodiments, includingsample algorithms, are described below to illustrate use in the contextof a Fourier-transform spectrometry instrument. These embodiments areprovided for illustrative purposes only and should not be consideredlimiting.

Example 1 Transfer Function of an Uncompensated System

The armature assembly and its flexure suspension form a lightly-dampedsecond-order rotational system. A torque balance at the armature yieldsthe following:

${Torque}_{Armature} = {\left. {{s^{2} \cdot \Theta \cdot J_{Armature}} + {s \cdot \Theta \cdot K_{Damping}} + {\Theta \cdot K_{Spring}}}\Rightarrow{Torque}_{Armature} \right. = {\Theta \cdot J_{Armature} \cdot \left( {{s^{2} \cdot s \cdot \frac{K_{Damping}}{J_{Armature}}} + \frac{K_{Spring}}{J_{Armature}}} \right)}}$

Solving for armature angle Θ:

$\Theta = {\left( \frac{{Torque}_{Armature}}{J_{Armature}} \right) \cdot \left( \frac{1}{{s^{2} \cdot s \cdot \frac{K_{Damping}}{J_{Armature}}} + \frac{K_{Spring}}{J_{Armature}}} \right)}$

The torque developed by the actuator is related to the current by theTorqueConstant_(Actuator):

Torque_(Armature)=ActuatorCurrent·TorqueConstant_(Actuator)

Therefore:

$\Theta = {\left( \frac{{ActuatorCurrent} \cdot {TorqueConstant}_{Actuator}}{J_{Armature}} \right) \cdot {\quad\left\lbrack \frac{1}{s^{2} + {s \cdot \left( \frac{K_{Damping}}{J_{Armature}} \right)} + \frac{K_{Spring}}{J_{Armature}}} \right\rbrack}}$

K_(UncompensatedPlant) was expressed as:

$K_{UncompensatedPlant} = \frac{\Theta}{ActuatorCurrent}$

Therefore:

$K_{UncompensatedPlant} = {\left( \frac{{TorqueConstant}_{Actuator}}{J_{Armature}} \right) \cdot \left\lbrack \frac{1}{s^{2} + {s \cdot \left( \frac{K_{Damping}}{J_{Armature}} \right)} + \frac{K_{Spring}}{J_{Armature}}} \right\rbrack}$$\mspace{20mu} {\omega_{n\; \_ \; {UncompensatedPlant}} = \sqrt{\frac{K_{Spring}}{J_{Armature}}}}$${2 \cdot \zeta_{UncompensatedPlant} \cdot \omega_{n}} = {\left. \frac{K_{Damping}}{J_{Armature}}\Rightarrow\zeta_{UncompensatedPlant} \right. = {\frac{K_{Damping}}{2} \cdot \sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}}$

Example 2 Developing Transfer Functions for Servo Components

Transfer functions were developed to define the forward branch of thecompensated plant loop:

${ForwardGain}_{CompensatedPlant} = \frac{\Theta_{Armature}}{e_{{Out}\; \_ \; {ErrorAmp}}}$Θ_(Armature) = e_(Out _ ErrorAmp) ⋅ Gm ⋅ K_(UncompensatedPlant)

Therefore:

ForewardGain_(CompensatedPlant) =Gm·K _(UncompensatedPlant)

The transconductance (Gm) amplifier was expressed as:

${Gm} = \frac{i_{Motor}}{e_{{Out}\; \_ \; {ErrorAmp}}}$

Therefore:

     ForwardGain_(CompensatedPlant) = Gm ⋅ K_(UncompensatedPlant)${ForwardGain}_{CompensatedPlant} = {{Gm} \cdot \left\lbrack {\left( \frac{{TorqueConstant}_{Actuator}}{J_{Armature}} \right) \cdot \left\lbrack \frac{1}{s^{2} + {s \cdot \left( \frac{K_{Damping}}{J_{Armature}} \right)} + \frac{K_{Spring}}{J_{Armature}}} \right\rbrack} \right\rbrack}$

Transfer functions were developed to express the feedback branch of thecompensated plant loop:

${FeedbackGain}_{CompensatedPlant} = \frac{e_{RateSummer\_ Out}}{\frac{}{t}\Theta_{Armature}}$e_(RateSummer_Out)  represents  the  angular  rate  at  the  armature.

The feedback variable e_(RateSummer) _(—) _(Out) is synthesized fromvoltages observed at the Top and the Bottom of the actuator. The voltageat the Bottom of the actuator coil is a function of the motor currentgiven by:

e _(MotorBottom) =i _(Motor) ·R _(Sense)

The voltage appearing at the top of the motor due to current from the Gmamplifier is:

e _(MotorTop) _(—) _(GmAmplifierCurrent) =i _(Motor)·(R _(Motor) +s·L_(Motor) +R _(Sense))

In addition to the above voltage, the back EMF constant of the motorgenerates a voltage proportional to armature motion:

$e_{MotorTop\_ BEMF} = {\left( {\frac{}{t}\Theta_{Armature}} \right) \cdot K_{BEMF}}$

The two voltages add yielding a voltage at the top of the motor givenby:

$e_{MotorTop} = {{i_{Motor} \cdot \left( {R_{Motor} + {s \cdot L_{Motor}} + R_{Sense}} \right)} + {\left( {\frac{}{t}\Theta_{Armature}} \right) \cdot K_{BEMF}}}$

The Lead Amplifier scales the voltage observed at the Motor Bottom suchthat the output of the Lead Amplifier is identical to that component ofthe voltage appearing at the Motor Top which is due to motor Current.The Rate Summer amplifier subtracts the output of the Lead Amplifierfrom the voltage appearing at the Motor Top. The output of the RateSummer represents only the motional voltage produced by armature motion,the effects of motor current having cancelled in the subtraction. Atransfer function for the lead amplifier was developed which satisfiesthe restraints and requirements of the system. For the effects of motorcurrent to cancel at the Rate Summer, the transfer function of the LeadAmplifier must satisfy the expression:

(e _(MotorBottom)·TransferFunction_(LeadAmplifier) =−e _(MotorTop) _(—)_(LockedArmature))

Therefore:

${TransferFunction}_{LeadAmplifier} = {{- 1} \cdot \frac{{i_{Motor} \cdot \left( {R_{Motor} + {s \cdot L_{Motor}} + R_{Sense}} \right)} + {(0) \cdot K_{BEMF}}}{i_{Motor} \cdot R_{Sense}}}$$\mspace{79mu} {{TransferFunction}_{LeadAmplifier} = {{- 1} \cdot \frac{R_{Motor} + {s \cdot L_{Motor}} + R_{Sense}}{R_{Sense}}}}$     TransferFunction_(LeadAmp) = Gain_(LeadAmp_0) ⋅ (s ⋅ τ_(LeadAmp_Zero_0) + 1)     When:${Gain}_{{LeadAmp\_}0} = {{{{- 1} \cdot \frac{R_{Sense} + R_{Motor}}{R_{Sense}}}\tau_{{LeadAmp\_ Zero}\_ 0}} = \frac{L_{Motor}}{R_{Sense} + R_{Motor}}}$

The Rate Summer subtracts the Lead Amplifier output from the voltagepresent at the Motor Top, yielding a signal which is a function ofArmature position alone. The Rate Summer scales this signal by thefactor K_(RateSummer). The output of the Rate Summer is:

e _(RateSummerOut)=(e _(MotorTop) +e _(LeadAmplfierOut))·K _(RateSummer)

Substituting the previous expressions for eMotorTop, eMotorBottom, andthe Lead Amplifier Transfer Function, the output of the Rate Summerbecomes:

$e_{RateSummerOut} = {\quad{{{\begin{bmatrix}{{\left\lbrack {{i_{Motor} \cdot \left( {R_{Motor} + {s \cdot L_{Motor}} + R_{Sense}} \right)} + {\left( {s \cdot \Theta_{Armature}} \right) \cdot K_{BEMF}}} \right\rbrack \mspace{14mu} \ldots} +} \\{\left( {i_{Motor} \cdot R_{Sense}} \right) \cdot \left( {{- 1} \cdot \frac{R_{Sense} + R_{Motor}}{R_{Sense}}} \right) \cdot \left( {{s \cdot \frac{L_{Motor}}{R_{Sense} + R_{Motor}}} + 1} \right)}\end{bmatrix} \cdot K_{RateSummer}}e_{RateSummerOut}} = {{K_{RateSummer} \cdot s \cdot \Theta_{Armature} \cdot {K_{BEMF}{units}}}\mspace{14mu} {are}\mspace{14mu} \left( \frac{Volt}{{Radians}_{Mechanical}} \right)}}}$

A transfer function was developed for an error amplifier:

e_(ErrorAmplifier_Out) = (e_(Command) − e_(RateSummer_Out)) ⋅ Gain_(ErrorAmplifier)${{where}{Gain}_{ErrorAmplifier}}:={1 \cdot \frac{volt}{volt}}$

Transfer function for the Open-Loop Gain of the compensated loop wasdetermined:

LoopGain_(CompensatedPlant) = ForwardGain_(CompensatedPlant) ⋅ FeedbackGain_(CompensatedPlant) ⋅ Gain_(ErrorAmplifier)${LoopGain}_{CompensatedPlant} = {\quad\left\lbrack {{Gm} \cdot \left\lbrack {\left( \frac{{TorqueConstant}_{Actuator}}{J_{Armature}} \right) \cdot \left. \quad\left\lbrack \frac{1}{s^{2} + {s \cdot \left( \frac{K_{Damping}}{J_{Armature}} \right)} + \frac{K_{Spring}}{J_{Armature}}} \right\rbrack \right\rbrack} \right\rbrack \cdot \left( {s \cdot K_{BEMF} \cdot K_{RateSummer}} \right) \cdot {Gain}_{ErrorAmplifier}} \right.}$

Transfer function for the Closed-Loop Gain of the compensated loop wasdetermined:

$\mspace{79mu} {{ClosedLoopGain}_{CompensatedPlant} = \frac{{Forward}\; {Gain}_{CompensatedPlant}}{1 + {LoopGain}_{CompensatedPlant}}}$${ClosedLoopGain}_{CompensatedPlant} = {{\left( \frac{{TorqueConstant}_{Actuator} \cdot {Gm}}{J_{Armature}} \right) \cdot \text{?}}\frac{1}{s^{2} + {s \cdot \left( \frac{\text{?}}{J_{Armature}} \right.}}}$${ClosedLoopGain}_{CompensatedPlant} = {\left( \frac{{TorqueConstant}_{Actuator} \cdot {Gm}}{J_{Armature}} \right) \cdot \frac{1}{{s^{2} + s}{{\cdot \left( {2 \cdot \xi_{CompensatedPlant} \cdot \omega_{n\_ CompensatedPlant}} \right)} + \omega_{n\_ CompensatedPlant}^{2}}}}$?indicates text missing or illegible when filed

The resonant frequencies of the compensated and uncompensated plants aresubstantially identical (e.g. identical):

$\mspace{79mu} {\omega_{n\_ CompensatedPlant} = \sqrt{\frac{K_{Spring}}{J_{Armature}}}}$${2 \cdot \zeta_{CompensatedPlant} \cdot \omega_{n}} = \frac{\begin{matrix}{K_{Damping} + {{Gm} \cdot {TorqueConstant}_{Actuator} \cdot}} \\{K_{BEMF} \cdot K_{RateSummer} \cdot {Gain}_{ErrorAmplifier}}\end{matrix}}{J_{Armature}}$$\zeta_{CompensatedPlant} = {\frac{\begin{matrix}{K_{Damping} + {{Gm} \cdot {TorqueConstant}_{Actuator} \cdot}} \\{K_{BEMF} \cdot K_{RateSummer} \cdot {Gain}_{ErrorAmplifier}}\end{matrix}}{2} \cdot \sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}$$\mspace{79mu} {\zeta_{UncompensatedPlant} = {\frac{K_{Damping}}{2} \cdot \sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}}$

The damping of the Compensated Plant is the sum of two terms: thedamping in the Uncompensated Plant (which may be inherent), plus a termdetermined by the loop gain:

$\zeta_{CompensatedPlant} = {\zeta_{UncompensatedPlant} = {\frac{\begin{matrix}{{Gm} \cdot {TorqueConstant}_{Actuator} \cdot K_{BEMF} \cdot} \\{K_{RateSummer} \cdot {Gain}_{ErrorAmplifier}}\end{matrix}}{2} \cdot \sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}}$

Example 3 Selection of Circuit Parameters to Achieve Performance Metrics

Circuit values were chosen to produce a closed-loop gain meeting thefollowing performance metrics which were deemed suitable for aninstrument (e.g. Fourier-transform spectrometry instrument):

${ClosedLoopGain}_{0{\_ CompensatedPlant}}:={{\frac{4.03049}{3} \cdot {\frac{\deg}{volt}{ClosedLoopGain}_{0{\_ CompensatedPlant}}}} = {1.343\frac{\deg}{volt}}}$     ζ_(CompensatedPlant) := 0.9     ω_(n_CompensatedPlant) = ω_(n_UncompensatedPlant)     Θ_(Armature_Max) := 4.03049 ⋅ deg      Θ_(Armature_Min) := −4.03049 ⋅ deg 

Observed uncompensated plant performance using the above circuit values:

$\omega_{n\_ UncompensatedPlant}:={235.3 \cdot \frac{rad}{\sec}}$${SpringConstant}_{{Flexure\_}1}:=\frac{0.0296 \cdot {lbf} \cdot {in}}{\deg}$

$\mspace{79mu} {{SpringConstant}_{{Flexure\_}2}:=\frac{0.0004 \cdot {lbf} \cdot {in}}{\deg}}$     ζ_(UncompensatedPlant) := 0.0239$K_{Spring}:={{\left( {{SpringConstant}_{{Flexure\_}1} + {SpringConstant}_{{Flexure\_}2}} \right)K_{Spring}} = {3.39 \times 10^{- 3}\mspace{14mu} \frac{N \cdot m}{\deg}}}$$\omega_{n}^{2} = {{\sqrt{\frac{K_{Spring}}{J_{Armature}}}J_{Armature}}:={{\frac{K_{Spring}}{\omega_{n\_ UncompensatedPlant}^{2}}J_{Armature}} = {3.508 \times 10^{- 6}\mspace{14mu} {{kg} \cdot m^{2}}}}}$

Observed actuator parameters using the above circuit values:

L_(Motor) := 6.019 ⋅ mH R_(Motor) := 30.0 ⋅ ohm${{Torque}\; {Constant}_{Actuator}}:={0.1649 \cdot \frac{N \cdot m}{amp}}$$K_{BEMF}:={0.748 \cdot \frac{volt}{\frac{Rad}{\sec}}}$Radius_(MirrorArm) := 0.4 ⋅ inK_(Θ _To_OPD) := 4 ⋅ Radius_(MirrorArm)

G_(m) required to satisfy Closed-Loop gain requirement:

${ClosedLoopGain}_{CompensatedPlant} = {\left( \frac{{TorqueConstant}_{Actuator} \cdot {Gm}_{0}}{J_{Armature}} \right) \cdot \frac{1}{s^{2} \cdot s \cdot \text{?}}}$?indicates text missing or illegible when filed

Therefore, at frequency=0:

${ClosedLoopGain}_{\; {0{\_ CompensatedPlant}}} = \left( \frac{{TorqueConstant}_{Actuator} \cdot {Gm}_{0}}{K_{Spring}} \right)$

-   -   Setting this equal to the desired value of Closed Loop Gain, and        solving for the required value of G_(m):

${Gm}_{0{\_ Required}}:={{\frac{{ClosedLoopGain}_{0{\_ CompensatedPlant}}}{{TorqueConstant}_{Actuator}} \cdot {K_{Spring}{Gm}_{0{\_ Required}}}} = {27.616\mspace{14mu} \frac{mAmp}{volt}}}$

Rate-Summer gain required to yield desired damping:

$Ϛ_{CompensatedPlant} = {Ϛ_{UncompensatedPlant} + {\frac{\begin{matrix}{{{Gm}_{0} \cdot {Torque}}\; {{Constant}_{Actuator} \cdot K_{{BEMF} \cdot}}} \\{K_{RateSummer} \cdot {Gain}_{ErrorAmplifier}}\end{matrix}}{2} \cdot \sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}}$

-   -   Setting this equal to the desired value of damping, and solving        for the required value of Rate Summer Gain:

$Ϛ_{CompensatedPlant} = {\xi_{UncompensatedPlant} + {\frac{\begin{matrix}{{{Gm}_{0} \cdot {Torque}}\; {{Constant}_{Actuator} \cdot K_{{BEMF} \cdot}}} \\{K_{RateSummer} \cdot {Gain}_{ErrorAmplifier}}\end{matrix}}{2} \cdot \sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}}$$K_{RateSummer\_ Required}:={{2 \cdot {\frac{\xi_{CompensatedPlant} - \xi_{UncompensatedPlant}}{\begin{matrix}{\left( \frac{1}{K_{Spring} \cdot J_{Armature}} \right)^{\frac{1}{2}} \cdot {Gm}_{0{\_ Required}} \cdot {TorqueConstant}_{Actuator} \cdot} \\{K_{BEMF} \cdot {Gain}_{ErrorAmplifier}}\end{matrix}}K_{RateSummer\_ Required}}} = 0.425}$

Example 4 Evaluation of Servo Transfer Functions

Compensated plant transfer functions were evaluated using the parametersdetermined in example 3.

Open-Loop transfer function (SEE FIG. 3):

$\mspace{79mu} {Ϛ_{UncompensatedPlant} = {\frac{K_{Damping}}{2} \cdot \sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}}$$\mspace{79mu} {K_{Damping}:={{\frac{2 \cdot \xi_{UncompensatedPlant}}{\sqrt{\frac{1}{K_{Spring} \cdot J_{Armature}}}}K_{Damping}} = {3.945 \times 10^{- 5}\mspace{14mu} \frac{N \cdot m}{\frac{rad}{\sec}}}}}$$\mspace{79mu} {{LoopGain}_{{CompensatedPlant}_{f}}:=\left\lbrack {{Gm}_{0{\_ Required}} \cdot {\quad{\left\lbrack {\left( \frac{{TorqueConstant}_{Actuator}}{J_{Armature}} \right) \cdot \left. \quad\left\lbrack \frac{1}{\left( s_{f} \right)^{2} + {s_{f} \cdot \left( \frac{K_{Damping}}{J_{Armature}} \right)} + \frac{K_{Spring}}{J_{Armature}}} \right\rbrack \right\rbrack} \right\rbrack \; \text{?}\text{?}\text{indicates text missing or illegible when filed}}}} \right.}$

Closed-Loop transfer function (SEE FIG. 4):

${ClosedLoopGain}_{{CompensatedPlant}_{f}}:={\left( \frac{{TorqueConstant}_{Actuator} \cdot {Gm}_{0{\_ Required}}}{J_{Armature}} \right) \cdot \text{?}}$${CurrentTransferFunction}_{{UncompensatedPlant}_{f}}:={{Gm}_{0{\_ Required}} \cdot \left\lbrack {{\left( \frac{{TorqueConstant}_{Actuator}}{J_{Armature}} \right) \cdot \text{?}}\text{?}\text{indicates text missing or illegible when filed}} \right.}$

Example 5 Circuit Synthesis

Circuits were developed to meet the performance requirements for theamplifiers.

A. Gm Amplifier Circuit Synthesis (SEE FIG. 5):

Driving Requirements:

${Gm}_{0{\_ Required}} = {27.616\mspace{14mu} \frac{mAmp}{volt}}$Θ_(Armature_Max) = 4.03049  deg  Θ_(Armature_Min) = −4.03049  deg ${freq}_{Gm}:={{10 \cdot {\frac{\omega_{n\_ UncompensatedPlant}}{2 \cdot \pi}{freq}_{Gm}}} = {374.492\frac{1}{s}}}$V_(dd) := 5 ⋅ volt V_(Out_Amp_Max) := V_(dd) − 1 ⋅ voltV_(ee) := −5 ⋅ volt V_(Out_Amp_Min) := V_(ee) − 1 ⋅ volt

Evaluation of R_(Sense):

$\frac{I_{Motor\_ Max} \cdot {TorqueConstant}_{Actuator}}{K_{Spring}} = {{\Theta_{Armature\_ Max}I_{Motor\_ Max}}:={{\frac{\Theta_{Armature\_ Max}}{{TorqueConstant}_{Actuator}} \cdot {K_{Spring}I_{Motor\_ Max}}} = {0.083\mspace{14mu} A}}}$

-   -   to avoid exceeding the output capability of the amplifier,        R_(Sense) is to satisfy the relationship:

${{\left( {I_{Motor\_ Max} + \frac{I_{Motor\_ Max} \cdot R_{Motor}}{R_{Sh}}} \right) \cdot R_{Sense\_ Max}} + {I_{Motor\_ Max} \cdot R_{Motor}}} = V_{{Out\_ Amp}{\_ Max}}$$R_{Sense\_ Max}:={{\frac{R_{Sh} \cdot \left( {V_{{Out\_ Amp}{\_ Max}} - {I_{Motor\_ Max} \cdot R_{Motor}}} \right)}{{R_{Sh} \cdot I_{Motor\_ Max}} - {I_{Motor\_ Max} \cdot R_{Motor}}}R_{Sense\_ Max}} = {16.675\mspace{14mu} \Omega}}$${PowerDissipation}_{\; {R\_ Sense}}:={{\left( \frac{I_{Motor\_ Max}}{\sqrt{12}} \right)^{2} \cdot {R_{Sense}{PowerDissipation}_{\; {R\_ Sense}}}} = {5.72 \times 10^{- 3}\mspace{14mu} {watt}}}$

Evaluation of Rln_Gm and R_(F) _(—) _(Gm):

$\mspace{79mu} {\gamma = \frac{R_{f}}{R_{in}}}$$\mspace{79mu} {I_{Motor} = {I_{AmpOut}\left( \frac{R_{Sh}}{R_{Sh} - R_{Motor}} \right)}}$$\mspace{79mu} {{{therfore}I_{AmpOut}} = {I_{Motor}\left( \frac{R_{Sh} - R_{Motor}}{R_{Sh}} \right)}}$     ande_(β) = I_(AmpOut) ⋅ R_(Sense)$\mspace{79mu} {{{substituting}e_{\beta}} = {I_{Motor} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense}}}$$\mspace{79mu} {{{At}\mspace{14mu} {the}\mspace{14mu} {inverting}\mspace{14mu} {input}\mspace{14mu} {of}\mspace{14mu} {the}\mspace{14mu} {{amplifier}\frac{e_{in}}{R_{in}}}} = \frac{e_{\beta}}{R_{f}}}$$\mspace{79mu} {{{therfore}\frac{e_{in}}{R_{in}}} = \frac{I_{Motor} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense}}{R_{f}}}$${{since}{Gm}_{0}} = {{\frac{I_{Motor}}{e_{in}}{Gm}_{0}} = {{\frac{R_{f}}{R_{in}} \cdot {\frac{R_{Sh}}{\left( {R_{Sh} + R_{Motor}} \right) \cdot R_{Sense}}{Gm}_{0}}} = {\gamma \cdot \frac{R_{Sh}}{\left( {R_{Sh} + R_{Motor}} \right) \cdot R_{Sense}}}}}$$\mspace{79mu} {\gamma:={{{Gm}_{0{\_ Required}}\text{?}\frac{\left( {R_{Sh} + R_{Motor}} \right) \cdot R_{Sense}}{R_{Sh}}{\text{?}\gamma}} = 0.304}}$$\mspace{79mu} {R_{In\_ Gm}:={{\frac{R_{F\_ Gm}}{\gamma}R_{In\_ Gm}} = {13896.1\mspace{14mu} \Omega}}}$?indicates text missing or illegible when filed

Evaluation of CE Gm:

-   -   The cutoff frequency of the Gm amplifier was expressed as:

$\mspace{79mu} {\text{?} = {{10 \cdot {\frac{\omega_{n\_ UncompensatedPlant}}{2 \cdot \pi}{freq}_{Gm}}} = {374.492\mspace{14mu} \frac{1}{s}}}}$?indicates text missing or illegible when filed

-   -   Therefore:

$\mspace{79mu} {\frac{1}{R_{f}C_{f}} = {10 \cdot \omega_{n\_ UncompensatedPlant}}}$$C_{F\_ Gm}:={{\frac{1}{R_{F\_ Gm} \cdot \left( {10 \cdot \omega_{n\_ UncompensatedPlant}} \right)}C_{F\_ Gm}} = {0.101\mspace{14mu} {\mu Farad}}}$     Standard  value:  C_(F_Gm) = 0.1  μFarad

Evaluation of G_(m) Amplifier Open-Loop Transfer Function:

-   -   The Open Loop gain of the transconductance amplifier is given by        the product:

LoopGain_(Gm) _(f) =VoltageTransferRatio_(α) _(—) _(To) _(—) _(β) _(f)·VoltageTransferRatio_(β) _(—) _(To) _(—) _(γ) _(f) ·A _(OpenLoop) _(—)_(OPA561) _(f)

-   -   Voltage divider from α to β:

$\mspace{79mu} {{VoltageTransferRatio}_{\; {{{\alpha\_}{To}}{\_\beta}}} = \frac{\beta}{\alpha}}$$\mspace{79mu} {\beta = {\alpha \cdot \frac{R_{Sense}}{R_{Sense} + Z_{\alpha \; \beta}}}}$$\mspace{79mu} {Z_{\alpha \; \beta} = \left\lbrack {\left( {R_{Motor} + {s \cdot L_{Motor}}} \right)^{- 1} + \frac{1}{R_{Sh}}} \right\rbrack^{- 1}}$${VoltageTransferRatio}_{\; {{{\alpha\_}{To}}{\_\beta}}} = \frac{R_{Sense}}{R_{Sense} + \frac{1}{\frac{1}{R_{Motor} + {s \cdot L_{Motor}}} + \frac{1}{R_{Sh}}}}$${VoltageTransferRatio}_{\; {{{\alpha\_}{To}}{\_\beta}_{f}}}:={R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +} \\{{R_{Sense} \cdot s_{f} \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}}$

-   -   Voltage divider from β to γ:

$\mspace{79mu} {{VoltageTransferRatio}_{\; {{{\beta\_}{To}}{\_\gamma}}} = \frac{\gamma}{\beta}}$$\mspace{79mu} {\gamma = {\beta \cdot \frac{R_{in}}{R_{in} + Z_{feedback}}}}$$\mspace{79mu} {Z_{feedback} = \left( {\frac{1}{R_{f}} + {s \cdot C_{f}}} \right)^{- 1}}$$\mspace{79mu} {{VoltageTransferRatio}_{\; {{{\beta\_}{To}}{\_\gamma}}} = \frac{R_{in}}{R_{in} + \frac{1}{\frac{1}{R_{f}} + {s \cdot C_{f}}}}}$${VoltageTransferRatio}_{\; {{{\beta\_}{To}}{\_\gamma}_{f}}}:=\frac{R_{In\_ Gm} + {R_{In\_ Gm} \cdot s_{f} \cdot C_{F\_ Gm} \cdot R_{F\_ Gm}}}{R_{In\_ Gm} + {R_{In\_ Gm} \cdot s_{f} \cdot C_{F\_ Gm} \cdot R_{F\_ Gm}} + R_{F\_ Gm}}$

-   -   OPA561 open loop gain:

A_(0_OPA 561) := 10⁵ freq_(3db_OPA 561) := 170 ⋅ Hz$A_{{OpenLoop\_ OPA}\; 561_{f}}:={A_{0{\_ OPA}\; 561} \cdot \frac{1}{{s_{f} \cdot \frac{1}{2 \cdot \pi \cdot {freq}_{3\; {db\_ OPA}\; 561}}} + 1}}$

-   -   The above quantities were substituted into the expression for        loop gain (SEE FIG. 6).

LoopGain_(Gm) _(f) :=A_(OpenLoop) _(—) _(OPA561) _(f)·VoltageTransferRatio_(α) _(—) _(To) _(—) _(β) _(f)·VoltageTransferRatio_(β) _(—) _(To) _(—) _(γ) _(f)

Evaluation of G_(m) Amplifier Closed-Loop Transfer Function:

${VoltageTransferRatio}_{\alpha \; {\_ To}{\_\beta}_{f}}:={R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +} \\{{R_{Sense} \cdot s_{f} \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}}$

-   -   Voltage appearing across the motor (SEE FIG. 7):

     Voltage_(Motor) = α − β     Voltage_(Motor) = α ⋅ (1 − VoltageTransferRatio_(α _To_β))$\mspace{34mu} {{{{su}{bstituting}}{Voltage}_{Motor}} = {\beta \cdot \left( \frac{1 - {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}}{{VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}} \right)}}$$\mspace{79mu} {I_{Motor} = \frac{{Voltage}_{Motor}}{R_{Motor} + {s \cdot L_{Motor}}}}$${{substituting}I_{Motor}} = {\beta \cdot \frac{1 - {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}}{{VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}} \cdot \left( {{R_{Motor} + s}{\cdot L_{Motor}}} \right)}}$${{solving}\mspace{14mu} {for}\mspace{14mu} {\beta \beta}} = {\frac{- I_{Motor}}{\left( {- 1} \right) + {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}} \cdot {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}} \cdot \left( {R_{Motor} + {s \cdot L_{Motor}}} \right)}$$\mspace{25mu} {{{summing}\mspace{14mu} {currents}\mspace{14mu} {at}\mspace{14mu} {the}\mspace{14mu} {op}\mspace{14mu} {{input}\frac{e_{in}}{R_{in}}}} = \frac{\beta}{\left( {\frac{1}{R_{F\_ Gm}} + {s \cdot C_{F\_ Gm}}} \right)^{- 1}}}$${{{substituting}{- 1}} \cdot \frac{e_{in}}{R_{In\_ Gm}}} = {{\frac{I_{Motor}}{\left( {- 1} \right) + {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}} \cdot {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}} \cdot \left( {R_{Motor} + {s \cdot L_{Motor}}} \right) \cdot \left( {\frac{1}{R_{F\_ Gm}} + {s \cdot C_{f}}} \right)}{\frac{- e_{in}}{\begin{matrix}{R_{In\_ Gm} \cdot {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}} \cdot} \\{\left( {R_{Motor} + {s \cdot L_{Motor}}} \right) \cdot \left( {1 + {s \cdot C_{F\_ Gm} \cdot R_{F\_ Gm}}} \right)}\end{matrix}} \cdot {\quad{{{\left\lbrack {\left( {- 1} \right) + {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}} \right\rbrack \cdot \text{?}}{solving}\mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {ratio}\mspace{14mu} {I_{Motor}/e_{in}}\mspace{20mu} \frac{I_{Motor}}{e_{in}}} = {\frac{\left( {- 1} \right)}{\begin{matrix}{R_{In\_ Gm} \cdot {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}} \cdot} \\{\left( {R_{Motor} + {s \cdot L_{Motor}}} \right) \cdot \left( {1 + {s \cdot C_{F\_ Gm} \cdot R_{F\_ Gm}}} \right)}\end{matrix}} \cdot {\quad{{{\left\lbrack {\left( {- 1} \right) + {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}} \right\rbrack \cdot \text{?}}{{therfore}{Gm}}} = {\frac{\left( {- 1} \right)}{\begin{matrix}{R_{In\_ Gm} \cdot {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}} \cdot} \\{\left( {R_{Motor} + {s \cdot L_{Motor}}} \right) \cdot \left( {1 + {s \cdot C_{F\_ Gm} \cdot R_{F\_ Gm}}} \right)}\end{matrix}} \cdot {\quad{{\left\lbrack {\left( {- 1} \right) + {VoltageTransferRatio}_{{{\alpha\_}{To}}{\_\beta}}} \right\rbrack \mspace{79mu} {substituting}\mspace{31mu} {the}\mspace{14mu} {Voltage}\mspace{14mu} {Transfer}\mspace{14mu} {Ratio}\mspace{14mu} {expressions}\text{}{Gm}} = {{{\frac{- 1}{R_{in} \cdot R_{Sense} \cdot \left( {R_{Sh} + R_{Motor} + {s \cdot L_{Motor}}} \right)} \cdot \frac{\begin{matrix}{R_{Sense}{{\cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +}} \\{{R_{Sense} \cdot s \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s \cdot L_{Motor}}}\end{matrix}}{\left( {R_{Motor} + {s \cdot L_{Motor}}} \right) \cdot \left( {1 + {{s \cdot C_{f}}R_{f}}} \right)}} \text{?}\mspace{79mu} {Gm}} = {{\left( \frac{R_{F\_ Gm} \cdot R_{Sh}}{R_{Sense}{{\cdot R_{In\_ Gm} \cdot R_{Sh}} + {R_{Sense} \cdot R_{In\_ Gm} \cdot R_{Motor}}}} \right)\text{?}\mspace{79mu} {simplify}\mspace{76mu} G_{m_{f}}}:={\quad{\left\lbrack \frac{R_{F\_ Gm} \cdot R_{Sh}}{R_{Sense} \cdot R_{In\_ Gm} \cdot \left( {R_{Sh} + R_{Motor}} \right)} \right\rbrack \cdot {\quad{\text{?}\text{?}\text{indicates text missing or illegible when filed}}}}}}}}}}}}}}}}}$

Verification of Closed-Loop Performance Requirements of G_(m) Amplifier:

${Low}\mspace{14mu} {frequency}\mspace{14mu} {transconductance}\mspace{14mu} {{gain}{\quad{{G_{m_{0}}} = {{27.411\mspace{14mu} \frac{mAmp}{volt}\mspace{14mu} {Gm}_{0{\_ Required}}} = {{27.616\mspace{14mu} \frac{mAmp}{volt}{Output}\mspace{14mu} {voltage}\mspace{14mu} {{excursion}e_{{Out\_ OPA}\; 561{\_ Max}}}}:={{{I_{Motor\_ Max} \cdot \left( R_{Motor} \right)} + {I_{Motor\_ Max} \cdot \left( {1 - \frac{R_{Motor}}{R_{Sh}}} \right) \cdot {R_{Sense}e_{{Out\_ OPA}\; 561{\_ Max}}}}} = {{3.396\mspace{14mu} V{Closed}\text{-}{Loop}\mspace{14mu} {{Bandwidth}G_{{m\_}3{db}}}}:={{\frac{G_{m_{0}}}{\sqrt{2}}G_{{m\_}3{db}}} = {{19.382\mspace{14mu} \frac{mAmp}{volt}{evaluating}\mspace{14mu} {the}\mspace{14mu} {Gm}\mspace{14mu} {function}\mspace{14mu} {at}\mspace{14mu} {the}\mspace{14mu} 3\mspace{14mu} {db}\mspace{14mu} {{frequency}{G_{m_{257}}}}} = {{19.51\mspace{14mu} {\frac{mAmp}{volt}{freq}_{257}}} = {371.535\mspace{14mu} {Hz}}}}}}}}}}\mspace{11mu}}}$

B. Lead Amplifier Circuit Synthesis (See FIG. 8):

Driving requirements provide that the product of the voltage at theMotor Low terminal and the Lead Amplifier Transfer Function is identicalto the voltage observed at the Motor High when the Armature is locked.The output of the Lead Amplifier has a phase shift of 180 degrees (i.e.,inversion).

The voltage at the Motor Bottom is:

$e_{MotorBottom} = {e_{\beta} = {I_{Motor} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense}}}$

The ratio of the voltage at the Motor Bottom to that at the Motor Topis:

${VoltageTransferRatio}_{\alpha \; {\_ To}{\_\beta}_{f}} = {R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +} \\{{R_{Sense} \cdot s_{f} \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}}$

Therefore:

$e_{{MotorTop}_{f}} = \frac{e_{{MotorBottom}_{f}}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +} \\{{R_{Sense} \cdot s_{f} \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}}$

-   -   Dividing by the voltage at the Motor Bottom yields an expression        for the required Lead Amplifier Closed Loop Transfer Function:

${ClosedLoopGain}_{LeadAmplifier} = \frac{1}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +} \\{{R_{Sense} \cdot s_{f} \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}}$${ClosedLoopGain}_{{LeadAmplifier\_ Required}_{f}}:={\quad{\left\lbrack \frac{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}}}{R_{Sense} \cdot \left( {R_{Sh} + R_{Motor}} \right)} \right\rbrack \text{?}\text{?}\text{indicates text missing or illegible when filed}}}$

The Closed Loop Transfer Function of the Lead Amplifier has the form:

${TransferFunction}_{LeadAmp\_ Required} = {{Gain}_{{LeadAmp\_}0} \cdot \frac{{s \cdot \tau_{{LeadAmp\_ Zero}\_ 1}} + 1}{{s \cdot \tau_{{LeadAmp\_ Pole}\_ 1}} + 1}}$${Gain}_{{LeadAmp\_}0}:={{{- 1} \cdot {\frac{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}}}{R_{Sense} \cdot \left( {R_{Sh} + R_{Motor}} \right)}{Gain}_{{LeadAmp\_}0}}} = {- 3.728}}$$\tau_{{LeadAmp\_ Zero}\_ 1}:={{L_{Motor} \cdot {\left( \frac{R_{Sense} + R_{{Sh}\;}}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}}} \right)\tau_{{LeadAmp\_ Zero}\_ 1}}} = {151.694\mspace{14mu} \mu \; {Sec}}}$$\tau_{{LeadAmp\_ Pole}\_ 1}:={{L_{Motor} \cdot {\left( \frac{1}{R_{Sh} + R_{Motor}} \right)\tau_{{LeadAmp\_ Pole}\_ 1}}} = {18.184\mspace{14mu} \mu \; {Sec}}}$

An expression for the Lead Amplifier Closed-Loop Response was developed:

$\mspace{79mu} {e_{Out} = {e_{In} \cdot \left( {- 1} \right) \cdot \frac{Z_{Feedback}}{Z_{In}}}}$$\mspace{79mu} {{ClosedLoopGain}_{LeadAmplifier} = \frac{e_{Out}}{e_{In}}}$$\mspace{79mu} {{ClosedLoopGain}_{LeadAmplifier} = {\left( {- 1} \right) \cdot \left( \frac{Z_{Feedback}}{Z_{In}} \right)}}$$Z_{Feedback} = {{\left( {\frac{1}{R_{f}} + {s \cdot C_{f}}} \right)^{- 1}\mspace{14mu} {{and}\mspace{14mu} Z_{In}}} = \left( {\frac{1}{R_{In}} + \frac{1}{R_{Stop} + \frac{1}{s \cdot C_{Lead}}}} \right)^{- 1}}$${ClosedLoopGain}_{LeadAmplifier} = \frac{{\left\lbrack {{\left( {- R_{f}} \right) \cdot R_{Stop} \cdot C_{Lead}} - {R_{f}{C_{Lead} \cdot R_{In}}}} \right\rbrack \cdot s} - R_{f}}{\begin{matrix}{{{R_{In} \cdot s^{2} \cdot C_{f}}R_{f}{R_{Stop} \cdot C_{Lead}}} +} \\{{\left( {{R_{In} \cdot R_{Stop} \cdot C_{Lead}} + {{R_{In} \cdot C_{f}}R_{f}}} \right) \cdot s} + R_{In}}\end{matrix}}$${ClosedLoopGain}_{LeadAmplifier} = {\frac{- R_{f}}{R_{In}} \cdot \frac{{\left( {R_{Stop} + R_{In}} \right) \cdot C_{Lead} \cdot s} + 1}{{{s^{2} \cdot C_{f}}R_{f}{R_{Stop} \cdot C_{Lead}}} + {s \cdot \left( {{R_{Stop} \cdot C_{Lead}} + {C_{f}R_{f}}} \right)} + 1}}$${ClosedLoopGain}_{LeadAmplifier} = {{- 1} \cdot \left( \frac{R_{f}}{R_{In}} \right) \cdot \left\lbrack \frac{{s \cdot \left\lbrack {\left( {R_{In} + R_{Stop}} \right) \cdot C_{Lead}} \right\rbrack} + 1}{\left( {{s \cdot R_{Stop} \cdot C_{Lead}} + 1} \right) \cdot \left( {{{s \cdot C_{f}}R_{f}} + 1} \right)} \right\rbrack}$

Therefore:

${ClosedLoopGain}_{LeadAmplifier} = {A_{0{\_ LeadAmplifier}} \cdot \left\lbrack \frac{{s \cdot \left( \tau_{{Zero\_}1{\_ LeadAmplifier}} \right)} + 1}{\left( {{s \cdot \tau_{{Pole\_}1{\_ LeadAmplifier}}} + 1} \right) \cdot \left( {{s \cdot \tau_{{Pole\_}2{\_ LeadAmplifier}}} + 1} \right)} \right\rbrack}$$\mspace{79mu} {A_{0{\_ LeadAmplifier}} = {{- 1} \cdot \frac{R_{F\_ LeadAmplifier}}{R_{In\_ LeadAmplifier}}}}$τ_(Zero_1_LeadAmplifier) = (R_(In_LeadAmplifier) + R_(Stop_LeadAmplifier)) ⋅ C_(Lead_LeadAmplifier)     τ_(Pole_1_LeadAmplifier) = R_(Stop_LeadAmplifier) ⋅ C_(Lead_LeadAmplifier)     τ_(Pole_2_LeadAmplifier) = R_(f_LeadAmplifier) ⋅ C_(f_LeadAmplifier)

-   -   In some embodiments, the frequency of the second pole is placed        above the zero to achieve the desired Lead-Lag response.        -   Synthesis of Pole 1:

$\mspace{79mu} {\tau_{{LeadAmp\_ Pole}\_ 1}:={L_{Motor} \cdot \left( \frac{1}{R_{Sh} + R_{Motor}} \right)}}$     τ_(Pole_1_LeadAmplifier) = R_(Stop_LeadAmplifier) ⋅ C_(Lead_LeadAmplifier)$\mspace{79mu} {{R_{Stop\_ LeadAmplifier} \cdot C_{Lead\_ LeadAmplifier}} = {L_{Motor} \cdot \left( \frac{1}{R_{Sh} + R_{Motor}} \right)}}$     C_(Lead_LeadAmplifier) := 0.02 ⋅ μFarad$R_{Stop\_ LeadAmplifier}:={{\frac{L_{Motor} \cdot \left( \frac{1}{R_{Sh} + R_{Motor}} \right)}{C_{Lead\_ LeadAmplifier}}R_{Stop\_ LeadAmplifier}} = {909.215\mspace{14mu} \Omega}}$     Standard  Value:  R_(Stop_LeadAmplifier) = 909 ⋅ ohm

-   -   -   Synthesis of Zero 1:

$\tau_{{Zero\_}1{\_ LeadAmplifier}}:={L_{Motor} \cdot \frac{R_{Sense} + R_{Sh}}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}}}}$τ_(Zero_1_LeadAmplifier) = (R_(In_LeadAmplifier) + R_(Stop_LeadAmplifier)) ⋅ C_(Lead_LeadAmplifier)$R_{In\_ LeadAmplifier}:={{{L_{Motor} \cdot \frac{R_{Sense} + R_{Sh}}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}}} \cdot \left( \frac{1}{C_{Lead\_ LeadAmplifier}} \right)} - {R_{Stop\_ LeadAmplifier}R_{In\_ LeadAmplifier}}} = {6675.72\mspace{14mu} \Omega}}$  Standard  Value:  R_(In_LeadAmplifier) = 6675.72  Ω

-   -   -   Synthesis of Gain₀:

$\mspace{79mu} {{Gain}_{{LeadAmp\_}0} = {{- 1} \cdot \frac{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}}}{R_{Sense} \cdot \left( {R_{Sh} + R_{Motor}} \right)}}}$$\mspace{79mu} {A_{0{\_ LeadAmplifier}} = {{- 1} \cdot \frac{R_{F\_ LeadAmplifier}}{R_{In\_ LeadAmplifier}}}}$$R_{F\_ LeadAmplifier}:={{R_{In\_ LeadAmplifier} \cdot {\left\lbrack \frac{{R_{{Sense}\;} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sh} \cdot R_{Motor}}}{R_{Sense} \cdot \left( {R_{Sh} + R_{Motor}} \right)} \right\rbrack R_{F\_ LeadAmplifier}}} = {24791.843\mspace{14mu} \Omega}}$     Standard  value:  R_(F_LeadAmplifier) = 24.9 ⋅ kOhm

-   -   -   Synthesis of Pole 2:

     τ_(Pole_2) := 0.1 ⋅ τ_(LeadAmp_Pole_1)     τ_(Pole_2_LeadAmplifier) = R_(F_LeadAmplifier) ⋅ C_(F_LeadAmplifier)$C_{F\_ LeadAmplifier}:={{\frac{\tau_{{Pole\_}2}}{R_{F\_ LeadAmplifier}}C_{F\_ LeadAmplifier}} = {73.029\mspace{14mu} {pFarad}}}$     Standard  value:  C_(F_LeadAmplifier) = 68 ⋅ pFarad

-   -   Evaluation of Lead Amplifier Closed-Loop Response by        substitution of values determined during Lead Amplifier        synthesis into the expression for the Lead amplifier Closed-Loop        Transfer Function (SEE FIG. 9).

${ClosedLoopGain}_{{LeadAmplifier}_{f}}:={\left( {{- 1} \cdot \frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}} \right) \cdot \text{?}}$?indicates text missing or illegible when filed

The response of the Lead amplifier was validated:

I _(Motor) _(—) _(Peak) _(f) :=I _(Motor) _(—) _(Max)

The voltage at the Motor Bottom when the motor is driven at I_(Motor)_(—) _(Peak):

$\mspace{79mu} {e_{{\beta\_}\; {Peak}_{f}}:={I_{{Motor}\; \_ \; {Peak}_{f}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot \text{?}}}$?indicates text missing or illegible when filed

-   -   Multiplying by the Lead Amplifier Closed Loop Transfer Function        yields an expression for the Locked-Armature output Voltage        assuming a one-amp motor current:

$e_{{Out}\; \_ \; {LeadAmp}\; \_ \; {Peak}_{f}}:={\left\lbrack {I_{{Motor}\; \_ \; {Peak}_{f}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense}} \right\rbrack \cdot {\quad\left\lbrack {\left( {{- 1}{\cdot \frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}}} \right) \cdot {\quad{\text{?}\text{?}\text{indicates text missing or illegible when filed}}}} \right.}}$

The voltage at the Motor Top under conditions of a One-amp motor currentis:

$e_{{MotorTop}\; \_ \; {Peak}_{f}}:=\frac{I_{{Motor}\; \_ \; {Peak}_{f}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +} \\{{R_{Sense} \cdot s_{f} \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}}$

-   -   Validation revealed that Lead Amplifier response was correct for        frequencies below approximately 30 kHz (SEE FIG. 10).

C. Rate Summer Amplifier Circuit Synthesis (See FIG. 11):

The desired output of the Rate Summer Amplifier, according to the servodesign is:

K _(RateSummer) _(—) _(Required)=0.425

-   -   An expression for the Rate Summer Closed-Loop Response was        developed. Since the gains applied to both inputs are        substantially, e.g. identical:

     R_(β) = R_(α) = R_(In _ RateSummer)${{Transfer}\mspace{14mu} {Function}_{RateSummer}} = {{- 1} \cdot \left( {Gain}_{0\; \_ \; {RateSummer}} \right) \cdot \left( \frac{1}{{s \cdot \tau_{{Pole}\; \_ \; {RateSummer}}} + 1} \right)}$     whereτ_(Pole _ RateSummer) = C_(f _ RateSummer) ⋅ R_(f _ RateSummer)$\mspace{79mu} {{Gain}_{0\; \_ \; {RateSummer}} = {{- 1} \cdot \left( \frac{R_{F\; \_ \; {RateSummer}}}{R_{{In}\; \_ \; {RateSummer}}} \right)}}$

Synthesis of Pole_(RateSummer):

τ_(Pole _ RateSummer) := 0.05 ⋅ τ_(LeadAmp _ Pole _ 1)C_(F _ RateSummer) := 100 ⋅ pFaradτ_(Pole _ RateSummer) = R_(F _ RateSummer) ⋅ C_(F _ RateSummer)${{therfore}R_{F\; \_ \; {RateSummer}}}:={{\frac{\tau_{{Pole}\; \_ \; {RateSummer}}}{C_{F\; \_ \; {RateSummer}}}\mspace{85mu} R_{F\; \_ \; {RateSummer}}} = {9092.145\Omega}}$Standard  value:  R_(F  RateSummer) = 9090 ⋅ ohm

Synthesis of Gain₀ _(—) _(Ratesummer):

K _(RateSummer) _(—) _(Required)=0.425

-   -   The subtraction required at the error amplifier was accomplished        by an inversion implemented at the Rate Summer, absorbing this        inversion into the Rate Summer design

$\mspace{20mu} {{Gain}_{0\; \_ \; {RateSummer}} = {{- 1} \cdot \frac{R_{F\; \_ \; {RateSummer}}}{R_{{In}\; \_ \; {RateSummer}}}}}$$R_{{In}\; \_ \; {RateSummer}}:={{\frac{R_{F\; \_ \; {RateSummer}}}{K_{{RateSummer}\; \_ \; {required}}}R_{{In}\; \_ \; {RateSummer}}} = {21410.051\Omega}}$  Standard  value:  R_(In _ RateSummer) = 21500 ⋅ ohm$\mspace{20mu} {K_{RateSummer}:={{\frac{R_{F\; \_ \; {RateSummer}}}{R_{{In}\; \_ \; {RateSummer}}}K_{RateSummer}} = 0.423}}$

D. The Feedback Branch Response to I_(Motor) and Θ_(Armature) wasEvaluated.

The output of the Rate Summer is the superposition of two functions: Thedesired signal due to the motional EMF produced by the moving armature,and an error term arising from the incomplete rejection of the voltageproduced by the motor current. The output of the Rate Summer is givenby:

     e_(Out _ RateSummer) = (e_(Out _ LeadAmplifier) + e_(MotorTop)) ⋅ K_(RateSummer)$e_{{Out}\; \_ \; {LeadAmplifier}} = {\left\lbrack {I_{Motor} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense}} \right\rbrack \cdot {\quad{{\left( {{- 1}\frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}} \right) \cdot \text{?}}\text{?}\text{indicates text missing or illegible when filed}}}}$

The component of the voltage appearing at the Motor Top due to motorcurrent is:

$e_{{MotorTop}\; \_ \; {LockedArmature}} = \frac{I_{{Motor}\;} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} +} \\{{R_{Sense} \cdot s \cdot L_{Motor}} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s \cdot L_{Motor}}}\end{matrix}}}$

The Motor Top voltage due to Armature motion is:

$e_{{MotorTop}\; \_ \; {Motional}} = {\left( {\frac{}{t}\Theta_{Armature}} \right) \cdot K_{BEMF} \cdot K_{RateSummer}}$

-   -   Substituting these terms into the expression for voltage at the        output of the Rate Summer:

$e_{{Out}\; \_ \; {RateSummer}} = {\begin{bmatrix}{I_{Motor} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot} \\{R_{Sense} \cdot K_{RateSummer}}\end{bmatrix} \cdot {\quad\left\lbrack {{{\left( {{- 1} \cdot \frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}} \right) \cdot \text{?}}\text{?}} + {\quad{\left\lbrack \frac{I_{Motor} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s \cdot L_{Motor}}}{\begin{matrix}{{R_{Sense} \cdot R_{Sh}} + {R_{Sense} \cdot R_{Motor}} + {R_{Sense} \cdot s \cdot L_{Motor}} +} \\{{R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s \cdot L_{Motor}}}\end{matrix}}} \right\rbrack + {\frac{}{t}{\Theta_{Armature} \cdot \text{?}}\text{?}\text{indicates text missing or illegible when filed}}}}} \right.}}$

At frequencies below mechanical resonance, the motor current may beexpressed as a function of armature angle:

$\mspace{20mu} {\left. \Rightarrow\Theta_{Armature} \right. = {I_{Motor} \cdot \frac{{TorqueConstant}_{Actuator}}{K_{Spring}}}}\mspace{14mu}$$\mspace{20mu} {\left. {therfore}\Rightarrow I_{Motor} \right. = {\Theta_{Armature} \cdot \frac{K_{Spring}}{{TorqueConstant}_{Actuator}}}}$$e_{{Out}\; \_ \; {RateSummer}} = {\left\lbrack {\Theta_{Armature} \cdot \frac{K_{Spring}}{{TorqueConstant}_{Actuator}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}} \right\rbrack \cdot \left\lbrack {{\left( {{- 1} \cdot \frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}} \right) \cdot \text{?}} + {\quad{\left\lbrack \frac{\Theta_{Armature} \cdot \frac{\left( K_{Spring} \right)}{\left( {TorqueConstant}_{Actuator} \right)} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{R_{Sense} + R_{Sh} + {R_{Sense} \cdot R_{Motor}} + {R_{Sense} \cdot s_{f} \cdot}} \\{L_{Motor} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}} \right\rbrack + {\frac{}{t}{\Theta_{Armature} \cdot K_{BEMF} \cdot \text{?}}\text{?}\text{indicates text missing or illegible when filed}}}}} \right.}$

-   -   The portion of the Rate Summer output voltage that is due to the        motional EMF produced by the armature constitutes the feedback        signal.

FeedbackSignal_(Motional)=Θ_(Armature) ·s·K _(BEMF) ·K _(RateSummer)

-   -   The portion of the Rate Summer output voltage that is due solely        to the Motor Current constitutes the feedback error, this is        given by:

${FeedbackError}_{I\; \_ \; {Motor}} = {\quad{\left\lbrack {\Theta_{Armature} \cdot \frac{K_{Spring}}{{TorqueConstant}_{Actuator}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}} \right\rbrack \cdot {\quad\left\lbrack {{\left( {{- 1} \cdot \frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}} \right) \cdot \text{?}} + {\quad{\quad{\left\lbrack \frac{\begin{matrix}{\Theta_{Armature} \cdot \frac{K_{Spring}}{{TorqueConstant}_{Actuator}} \cdot} \\{\left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}}\end{matrix}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s \cdot L_{Motor}}}{\begin{matrix}{R_{Sense} + R_{Sh} + {R_{Sense} \cdot R_{Motor}} + {R_{Sense} \cdot s \cdot}} \\{L_{Motor} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s \cdot L_{Motor}}}\end{matrix}}} \right\rbrack \text{?}\text{indicates text missing or illegible when filed}}}}} \right.}}}$

Taking the ration of the feedback signal and the feedback error yieldsan expression for the SNR of the Feedback signal (SEE FIG. 12)

${SNR}_{{RateFeedback}_{f}} = \left\lbrack {{\frac{s_{f} \cdot K_{BEMF} \cdot K_{RateSummer}}{\begin{bmatrix}{\frac{K_{Spring}}{{TorqueConstant}_{Acutator}} \cdot} \\{\left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}}\end{bmatrix} \cdot {\quad\left\lbrack {\left( {{- 1} \cdot \frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}} \right) \cdot \left\lbrack \text{?} \right.} \right.}}\text{?}} + {\quad{\left\lbrack \frac{\frac{K_{Spring}}{{TorqueConstant}_{Actuator}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{R_{Sense} + R_{Sh} + {R_{Sense} \cdot R_{Motor}} + {R_{Sense} \cdot s_{f} \cdot}} \\{L_{Motor} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}} \right\rbrack \text{?}\text{indicates text missing or illegible when filed}}}} \right.$

E. Closed-Loop Transfer Function of Synthesized Circuits were Comparedto the Required Response, Assuming Standard-Value Components (See FIG.13).

${ForwardGain} = \frac{\Theta_{Armature}}{e_{{Out}\; \_ \; {RateSummer}}}$ForwardGain_(Synthesis) = G_(m) ⋅ K_(UncompensatedPlant)${FeedbackGain}_{Synthesis} = \frac{e_{{Out}\; \_ \; {RateSummer}}}{\Theta_{Armature}}$${ForwardGain}_{f} = {\left\lbrack \frac{R_{F\; \_ \; {Gm}} \cdot R_{Sh}}{R_{Sense} \cdot R_{{In}\; \_ \; {Gm}} \cdot \left( {R_{Sh} + R_{Motor}} \right)} \right\rbrack \cdot {\quad\left\lbrack {{\frac{1}{{\left( s_{f} \right)^{2} \cdot \left( \frac{R_{F\; \_ \; {Gm}} \cdot L_{Motor} \cdot C_{F\; \_ \; {Gm}}}{R_{Sh} + R_{Motor}} \right)} + {s_{f}\left( \text{?} \right.}}\text{?}{FeedbackGain}_{f}}:={\quad{\quad{\quad\left\lbrack {{\begin{matrix}{\begin{bmatrix}{\frac{K_{Spring}}{{TorqueConstant}_{Actuator}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot} \\{R_{Sense} \cdot K_{RateSummer}}\end{bmatrix} \cdot {\quad{\quad\left\lbrack {\left( {{- 1} \cdot \frac{R_{F\; \_ \; {LeadAmplifier}}}{R_{{In}\; \_ \; {LeadAmplifier}}}} \right) \cdot \left\lbrack {{\frac{s_{f} \cdot \left\lbrack \left( \text{?} \right. \right.}{s_{f} \cdot \left( {R_{{Stop}\; \_ \; {LeadAmplifier}} \cdot \text{?}} \right.}\text{?}} +} \right.} \right.}}} \\{\quad{\left\lbrack \frac{\frac{K_{Spring}}{{TorqueConstant}_{Actuator}} \cdot \left( \frac{R_{Sh} + R_{Motor}}{R_{Sh}} \right) \cdot R_{Sense} \cdot K_{RateSummer}}{R_{Sense} \cdot \frac{R_{Sh} + R_{Motor} + {s_{f} \cdot L_{Motor}}}{\begin{matrix}{R_{Sense} + R_{Sh} + {R_{Sense} \cdot R_{Motor}} + {R_{Sense} \cdot s_{f} \cdot}} \\{L_{Motor} + {R_{Sh} \cdot R_{Motor}} + {R_{Sh} \cdot s_{f} \cdot L_{Motor}}}\end{matrix}}} \right\rbrack + {s_{f} \cdot K_{BEMF} \cdot K_{RateSummer}}}}\end{matrix}{ClosedLoopTransferFunction}_{{CircuitSynthesis}_{f}}}:={\frac{{ForwardGain}_{f}}{1 + {{ForwardGain}_{f} \cdot {FeedbackGain}_{f}}}\text{?}\text{indicates text missing or illegible when filed}}} \right.}}}} \right.}}$

Any publications and patents mentioned in the present application areherein incorporated by reference. Various modification and variation ofthe described methods and compositions will be apparent to those skilledin the art without departing from the scope and spirit of the invention.Although specific embodiments have been described, it should beunderstood that the claims should not be unduly limited to such specificembodiments. Indeed, various modifications of the described modes andembodiments that are obvious to those skilled in the relevant fields areintended to be within the scope of the following claims. Moreover, whilethe techniques, systems, methods and approaches have been described withrespect to specific devices, it is to be apparent that a variety ofsystems and devices may benefit from the methods, devices, and systemsdescribed herein. Examples include, but are not limited to optical basedmedia devices, precision manufacturing devices, optical computingsystems, optical measuring devices, line-of-sight communication systems,optical communication devices, and so forth.

1. A method of electrically-augmented damping of an actuator,comprising: subtracting a rate-feedback voltage from an actuator commandvoltage, wherein the rate feedback voltage is derived from measurementof voltage across the actuator and a current flowing through theactuator.
 2. The method of claim 1, wherein the voltage across theactuator comprises a component due to flow of current through impedancesof the actuator, and a component due to movement of an armature includedin the actuator.
 3. The method of claim 2, wherein the rate-feedbackvoltage is derived by subtracting from the voltage across the actuator avoltage equal to the component due to the flow of current through theimpedances of the actuator.
 4. The method of claim 3, wherein thevoltage equal to the component due to the flow of current through theimpedances of the actuator is derived by scaling voltage appearing at aterminal β of the actuator.
 5. The method of claim 4, wherein thevoltage appearing at the terminal β of the actuator is a functionsubstantially of the current flowing through the actuator. 6.-7.(canceled)
 8. A device including circuitry for providingelectrically-augmented damping of a voice-coil actuator, comprising: a)a transconductance amplifer A1 configured to provide an output that is afunction of: i) command voltage, and ii) a rate-feedback voltage; b) avoice-coil actuator, including: i) a permanent magnet, ii) a terminal α,iii) a terminal β, and iv) an armature configured to carry currentthrough a magnetic field associated with the permanent magnet; c) anamplifier A2 configured to provide an output that is a function ofvoltage appearing at the terminal β of the voice-coil actuator; and d)an amplifier A3 configured to provide an output that is a function of:i) voltage appearing at the terminal α of the voice-coil actuator, andii) a voltage proportional to the voltage appearing at the terminal α ofthe voice-coil actuator due to a flow of current through the voice-coilactuator.
 9. The device of claim 8, wherein the device and voice-coilactuator are configured to cause the output of the amplifier A1 to flowinto the voice-coil actuator.
 10. The device of claim 8, wherein thevoltage appearing at the terminal β of the voice-coil actuator is afunction of the flow of current through the voice-coil actuator.
 11. Thedevice of claim 8, wherein the voltage appearing at the terminal α ofthe voice-coil actuator is a function of: i) the flow of current throughthe voice-coil actuator, and ii) the movement of the armature throughthe magnetic field.
 12. The device of claim 8, wherein the amplifier A2is configured to scale and invert the voltage appearing at the terminalβ of the voice-coil actuator to produce a voltage equal in magnitude to,but 180 degrees out of phase with, the voltage across appearing at theterminal α of the voice-coil actuator due to the flow of current throughthe voice-coil actuator.
 13. The device of claim 8, wherein theamplifier A3 is configured to provide an output that is the sum of theoutput of amplifier A2 and the voltage appearing at the terminal α ofthe voice-coil actuator.
 14. The device of claim 13, wherein theamplifier A3 is configured to provide an output that is equal to thevoltage resulting from the movement of the armature through the magneticfield.
 15. The device of claim 14, wherein the amplifier A3 isconfigured to output the rate-feedback voltage.
 16. A device comprisingan actuator comprising a system for electrically-augmented damping. 17.The device of claim 16, wherein a rate-feedback voltage is subtractedfrom an actuator command voltage to effectively damp the actuator. 18.The device of claim 17, wherein the rate feedback voltage is derivedfrom measurement of voltage across the actuator and current flowingthrough the actuator. 19.-20. (canceled)
 21. A Fourier-transformspectrometer comprising an optical path-length modulator utilizing anactuator comprising a system for electrically-augmented damping.
 22. TheFourier-transform spectrometer of claim 21, wherein a rate-feedbackvoltage is subtracted from the actuator command voltage to effectivelydamp the actuator.
 23. The Fourier-transform spectrometer of claim 22,wherein the rate feedback voltage is derived from measurement of voltageacross the actuator and current flowing through the actuator. 24.-25.(canceled)
 26. The Fourier-transform spectrometer of claim 21, whereinreduced performance anomalies for the actuator provides a reduction inerrors in time-domain interferograms, a reduction in errors infrequency-domain spectrum, and improved spectrometer performance incomparison to a Fourier-transform spectrometer that does not implementelectrically-augmented damping.